Mathematics · Cheatsheet

Advanced Complex Numbers (HL)

Chapter 1 · Euler's Formula & Identity

📋 Reference · always available

Three equivalent forms

z = a + bi = r(\cos\theta + i\sin\theta) = r\,e^{i\theta}

Cartesian for +/−, polar for ×/÷/powers, exponential most compact.

Euler's formula

e^{i\theta} = \cos\theta + i\sin\theta

Derived by substituting

x = i\theta

into the Maclaurin series of

e^x

and grouping real and imaginary parts.

Euler's identity

e^{i\pi} + 1 = 0

Special case at

\theta = \pi

. Uses

e, i, \pi, 1, 0

each once.

Four quadrant values

e^{i \cdot 0} = 1,\; e^{i\pi/2} = i,\; e^{i\pi} = -1,\; e^{i \cdot 3\pi/2} = -i

Multiplication rule

e^{i\theta_1} \cdot e^{i\theta_2} = e^{i(\theta_1 + \theta_2)}

Imaginary exponents obey the same exponent law as real ones.

Common trap

Track every power of

i

during Maclaurin substitution —

(i\theta)^4 = +\theta^4

(even power flips sign back to positive).